CDMA Uplink Power Control as a Noncooperative Game - CiteSeerX

Wireless Networks 8, 659–670, 2002  2002 Kluwer Academic Publishers. Manufactured in The Netherlands.

CDMA Uplink Power Control as a Noncooperative Game ∗ and R. SRIKANT TANSU ALPCAN, TAMER BASAR ¸

Coordinated Science Laboratory, University of Illinois, 1308 West Main Street, Urbana, IL 61801, USA

EITAN ALTMAN INRIA, B.P. 93, 06902 Sophia Antipolis cedex, France, and CESIMO, Facultad de Ingenieria, Universidad de Los Andes, Merida, Venezuela

Abstract. We present a game-theoretic treatment of distributed power control in CDMA wireless systems. We make use of the conceptual framework of noncooperative game theory to obtain a distributed and market-based control mechanism. Thus, we address not only the power control problem, but also pricing and allocation of a single resource among several users. A cost function is introduced as the difference between the pricing and utility functions, and the existence of a unique Nash equilibrium is established. In addition, two update algorithms, namely, parallel update and random update, are shown to be globally stable under specific conditions. Convergence properties and robustness of each algorithm are also studied through extensive simulations. Keywords: power control, CDMA, game theory, pricing, resource allocation

1. Introduction In wireless communication systems, mobile users respond to the time-varying nature of the channel, described using shortterm and long-term fading phenomena [9], by regulating their transmitter powers. Specifically, in a code division multiple access (CDMA) system, where signals of other users can be modeled as interfering noise signals, the major goal of this regulation is to achieve a certain signal-to-interference (SIR) ratio regardless of channel conditions while minimizing the interference due to user transmit power level. Hence, there are two major reasons for a user to exercise power control: the first one is the limit on the battery energy available to the mobile, and the second reason is the increase in capacity, which can be achieved by minimizing the interference. Power control in CDMA systems are in either open-loop or closed-loop form. In open-loop power control, the mobile regulates its transmitted power inversely proportional to the received power. In closed-loop power control, on the other hand, commands are transmitted to the mobile over the downlink to increase or decrease its uplink power [12, p. 182]. A specific proposal to implement distributed power control made by Yates [13] relies on each user updating its power based on the total received power at the base station. It has been shown in [13] that the resulting distributed power control algorithm converges under a wide variety of interference models. Another distributed power control scheme has been introduced in [7], which is adaptive and uses local measurements of the mean and the variance of the interference. The authors have shown that this algorithm is convergent provided that a certain condition is satisfied. Game theory provides a natural framework for developing pricing mechanisms of direct relevance to the power control ∗ Corresponding author.

problem in wireless networks. In such networks, the users behave noncooperatively, i.e., each user attempts to minimize its own cost function (or maximize its utility function) in response to the actions of the other users. This makes the use of noncooperative game theory [1] for uplink power control most appropriate, with the relevant solution concept being the noncooperative Nash equilibrium. In this approach, a noncooperative network game is defined where each user attempts to minimize a specific cost function by adjusting his transmission power, with the remaining users’ power levels fixed. The main advantage of this approach is that it not only leads to distributed control as in [13], but also naturally suggests pricing schemes, as we will see in this paper. Possible utility functions in a game theoretical framework, and their properties for both voice and data sources have been investigated in detail in [4], which formulates a class of utility functions that also account for forward error control, and shows the existence of a Nash equilibrium and the uniqueness of an optimal response. One interesting feature of this framework is that it provides utility functions for wireless data transmission, where power control directly affects the capacity of mobiles’ data transmission rates. Falomari et al. [4] also propose a linear pricing scheme in order to achieve a Pareto improvement in the utilities of mobiles. In an earlier study [10], Nash equilibria achieved under a pricing scheme have been characterized by using supermodularity. It has been shown that a noncooperative power control game with a pricing scheme is superior to one without pricing. One deficiency of this game setup, however, is that it does not guarantee social optimality for the equilibrium points. Linear and exponential utility functions based on carrier (signal)-to-interference ratio are also proposed in [5]. The existence of a Nash equilibrium is shown under some assumptions on the utility functions, and an algorithm for solving the noncooperative power control game is suggested.

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In this paper, we propose a power control game similar to the ones in [4,5]. In the model we adopt, however, we use a cost function defined as the difference between a linear pricing scheme proportional to transmitted power, and a logarithmic, strictly concave utility function based on SIR of the mobile. Furthermore, the utility function is made user-specific by multiplying it with a utility parameter reflecting the individual user preferences. We then rigorously prove the existence and uniqueness of a Nash equilibrium. We also investigate possible boundary equilibrium solutions, and hence derive a quantitative criterion for admission control. As in [13], one way of extending the model is to include certain SIR constraints. As an alternative, we suggest a pricing strategy to meet the given constraints, and analyze the relation between price, SIR, and user preferences as reflected by the utility parameter. Thus, we address not only the power control problem, but also pricing and allocation of a single resource among several users. Furthermore, we study different pricing strategies, and obtain a distributed and market-based power control mechanism. Finally, under a sufficient condition we prove the convergence of two algorithms, parallel update (PUA) and random update (RUA), to the unique Nash equilibrium. In order to illustrate the convergence, stability and robustness of the update algorithms, we use extensive simulations using MATLAB. Moreover, we study the effect of the various parameters of the model, especially different pricing schemes. In order for the simulations to capture realistic scenarios, we introduce feedback delay and modeling disturbances, where the latter is caused by variations in the number and location of users in the network. The next section describes the model adopted and the cost function. In section 3, we prove the existence and uniqueness of the Nash equilibrium. We present update algorithms for mobiles in section 4, whereas section 5 introduces different pricing strategies at the base station. The simulation results are given in section 6, which is followed by the concluding remarks of section 7.

2. The model and cost function We describe here the simple model adopted in this paper for a single cell CDMA system with up to M users. The number of users is limited under an admission control scheme that ensures the minimum necessary SIR for each user in the cell. For the ith user, we define the cost function Ji as the difference between the utility function of the user and its pricing function, Ji = Pi − Ui . The utility function, Ui , is chosen as a logarithmic function of the ith user’s SIR, which we denote by γi . This utility function can be interpreted as being proportional to the Shannon capacity [9,11] for user i, if we make the simplifying assumption that the noise plus the interference of all other users constitute an independent Gaussian noise. This means that this part of the utility is simply linear in the throughput that can be achieved (or approached) by user i using an appropriate coding, as a function of its transmission power. This logarithmic function is further weighted by

ALPCAN ET AL.

a user-specific utility parameter, ui > 0, to capture the user’s level of “desire” for SIR. The pricing function defines the instantaneous “price” a user pays for using a specific amount of power that causes interference in the system. It is a linear function of pi , the power level of the user. Accordingly, the cost function of the ith user is defined as Ji (pi , p−i ) = λi pi − ui ln(1 + γi ),

pi
0 ∀i,

(2.1)

where p−i denotes the vector of power levels of all users except the ith one, and γi is the SIR function for user i, given by γi = L


hi pi . 2 h j =i j pj + σ

(2.2)

Here, L = W/R is the spreading gain of the CDMA system, where W is the chip rate and R is the total rate; we assume throughout that L > 1. The parameter hj , 0 < hj < 1, is the channel gain from user j to the base station in the cell, and σ 2 > 0 is the interference. For notational convenience, let us denote the ith user’s power level received at the
base station as yi := hi pi , introduce the quantity y −i := j =i yj , and further define a user specific parameter (ai ) for the ith user as ai :=

ui hi σ2 . − λi L

(2.3)

3. Existence and uniqueness of Nash equilibrium The ith user’s optimization problem is to minimize its cost, given the sum of powers of other users as received at the base station, y −i , and noise. The nonnegativity of the power vector (pi
0 ∀i) is an inherent physical constraint of the model. Taking the derivative of the cost function (2.1) with respect to pi , we obtain the first-order necessary condition: Lui hi ∂Ji (pi , p−i ) = λi −

0. (3.1) 2 ∂pi h p j j + Lhi pi + σ j =i In the case of a positive inner solution, (3.1) holds with equality. It is easy to see that the second derivative is also positive, and hence the inner solution, if it exists, is the unique point minimizing the cost function. The boundary solution, pi = 0, is the other possible optimal point for the constrained optimization problem. If the user’s cost function, Ji (pi , p−i ), attains its minimum for a power value less than zero, pi,min < 0, the optimal solution will be the boundary point. Solving equation (3.1) and invoking the positivity constraint pi
0, we obtain the reaction function, i , of the ith user: pi = i (y −i , ai )    1 1   ai − y −i , L = hi   0,

if y −i  Lai ,

(3.2)

else.

The reaction function is the optimal response of the user to the varying parameters in the model. It depends not only on the user-specific parameters, like ui , λi , and hi , but also on

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Theorem 3.1. In the power game just defined (with M users), let the indexing be done such that ai < aj ⇒ i > j , with the ordering picked arbitrarily if ai = aj . Let M ∗  M be the  for which the following condition is satislargest integer M fied:  M

aM 

Figure 1. A simplified block diagram of the system.

the network parameter, L, and total power level received at
y the base station, M j =1 j . Actually, (3.2) shows dependence only on y −i , but adding (−pi /L) to both sides, and dividing both sides by (1 − 1/L), one can express
the response of the ith user as a function of the quantity M j =1 yj . The base station provides the user with total received power level using the downlink. If the frequency of user updates is sufficiently high, this can be done incrementally in order to decrease the overhead to the system. A simplified block diagram of the system is shown in figure 1. Similar to the transmission control protocol (TCP) in the Internet [8], there is an inherent feedback mechanism here, built into the reaction function of the user. In this model, the total received power at the base station provides the user with information about the “demand” in the network, which is comparable to congestion in case of the TCP. However, one major difference is that here the reaction function itself takes the place of the window based algorithms in the TCP. In order for the ith mobile to be “active”, or pi > 0 the following conditions from (3.2) have to hold: ai > 0 and y −i  Lai . An intuitive interpretation for these conditions is the following: if the price, λi , is set too high for a mobile, the mobile prefers not to transmit at all, depending on its channel gain and utility parameter, and the spreading gain and interference level. For any equilibrium solution, the set of fixed point equations can be written in matrix form by exploiting the linearity of (3.2). In case of a boundary solution, the rows and columns corresponding to users with zero equilibrium power are deleted, and the equation below involves only the users with positive powers. Hence, we have (assuming here that all M users have positive power levels at equilibrium):   hM h2 h3 ··· 1      Lh1 Lh1 Lh1   p1∗  c1   h1 h h 3 M  ∗     1 ···  p2   c2   Lh Lh2 Lh2  2  =    .   .   . .. ..  ..   ..   ..   . .  . . .   p∗  cM   h1 M h2 hM−1 ··· 1 LhM LhM LhM ⇔

Ap∗ = c,

(3.3)

where the variable ci is defined as ci = ai / hi . Note that ci > 0 if ai is positive.

 1 > ai .  (L + M − 1) i=1

(3.4)

Then, the power game admits a unique Nash equilibrium (NE), which has the property that users M ∗ + 1, . . . , M have zero power levels, pj∗ = 0, j
M ∗ + 1. The equilibrium power levels of the first M ∗ users are obtained uniquely from (3.3) with M replaced by M ∗ , and are given by     1 1 L ∗ aj , pi = ai − hi L − 1 L + M∗ − 1 j ∈M∗   i ∈ M∗ := 1, 2, . . . , M ∗ . (3.5)  for which (3.4) is satisfied, then the NE If there is no M solution is again unique, but assigns zero power level to all M users. Proof. We first state and prove the following lemma, which will be useful in the proof of the theorem.  = M,  it is Lemma 3.2. If condition (3.4) is satisfied for M  such that 1  M  < M.  also satisfied for all M  = M;  then Proof. Suppose that condition (3.4) holds for M   we argue that it also holds for M = M − 1. Substituting  for M,  we rewrite it as in (3.4) M 

M−2     − 2 a − a > L+M ai . M M−1 i=1


aM Due to the indexing of users, we have aM−1   . Substituting aM−1 for aM   above, we obtain 



M−2    − 3 a > ai . L+M M−1 i=1

 = M  − 1. The proof then Hence, (3.4) is satisfied for M  follows by induction on M.  Returning to the proof of the theorem, we first show that the matrix A in equation (3.3) is full rank and hence invertible, and thereby the solution to (3.3), p∗ , is unique. Then we show that the solution is strictly positive (i.e., pi∗ > 0 ∀i ∈ M)  = M. Finally, if, and only if, condition (3.4) is satisfied for M we relax condition and allow for boundary solutions, and conclude the proof by proving the uniqueness of the boundary solution. In order for the matrix A in (3.3) to be full rank and hence invertible, there should not exist a nonzero vector

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q = (q1 q2 . . . qM )T =  0 such that Aq = 0. This equation can be written as  Lhi qi + hj qj = 0 ∀i j =i

⇒ (L − 1)hi qi +

M 

hj qj = 0

∀i.

(3.6)

Summing up this set of equations over all users (i = 1, . . . , M), we arrive at  M   (L − 1 + M) hj qj = 0. j =1

The
M term L − 1 + M above is nonzero, and hence, the sum j =1 hj qj has to be zero. Since the channel gains are strictly positive, hi > 0 ∀i, and L > 1, it follows from (3.6) that qi = 0 ∀i. Accordingly, the matrix A is full rank and hence invertible, which leads to a unique solution for equation (3.3). Simple manipulations lead to (3.5), with M ∗ = M, for this unique solution. If the NE exists and is strictly positive, then (3.3) has to have a unique positive solution, which we already know is given by (3.5). Hence, (3.5) has to be positive, which is precisely condition (3.4) in view of also the indexing of the users.  = M, then we obtain On the other hand, if (3.4) holds for M from (3.5) that the equilibrium power level of each user is strictly positive. The existence and uniqueness of the NE follows from (3.3). We thus conclude that condition (3.4) with  = M is both necessary and sufficient for the existence of M a unique positive Nash equilibrium. To complete the proof for the case M ∗ = M, possible boundary solutions need to be investigated to conclude the uniqueness of the inner Nash equilibrium. We have to show  of M  that there cannot be another NE, with a subset M users transmitting with positive power, and the remaining  users having zero power level. In this case, the reM−M  is given by (3.5) active power level of the ith mobile, i ∈ M, ∗  with M = M.  in order for the zero power For any ith mobile, i ∈ / M, level to be part of a Nash equilibrium, condition (3.7)

should fail according to the reaction function (3.2) of the mobile. Summing up the equilibrium power levels as received  users with positive power levels by the base station of M  results in (from (3.5) with M ∗ = M)  1 1  yj = aj .  L L+M −1   j ∈M

(3.8)

j ∈M

Substituting in (3.7) the expression (3.8) for y −i yields ai


 1 aj . −1 L+M  j ∈M

 M

ai


j =1

y −i  Lai

On the other hand, from lemma 3.2, and (3.4), we have  + 1} the followfor any ith user in the indexed set {1, . . . , M ing:

(3.9)

 1 aj .  L + M − 1 j =1

(3.10)


 Also, from the indexing of the users it follows that M j =1 aj

a . Using this in (3.10), we see that inequality (3.9)  j j ∈M  + 1}, regardis satisfied for any ith user, i ∈ {1, . . . , M  We note that there less of the choice of the subset M. exists at least one user belonging to the set {1, . . . ,  Thus, the power of that mobile  + 1}, but not the subset M. M must be positive, and hence the boundary solution cannot be a  Nash equilibrium. As this argument is valid for any subset M, all boundary solutions fail similarly for being an equilibrium, including the trivial solution, the origin. We thus conclude that the inner Nash equilibrium is unique. This completes the proof for the case M ∗ = M. If M ∗ < M in condition (3.4), then the equilibrium (whenever it exists) will clearly be a boundary point. If condition (3.4) fails for users M ∗ + 1, . . . , M where users are indexed as described in theorem 3.1, then these users use zero power in the equilibrium. Hence, for any ith user among M ∗ + 1, . . . , M, condition (3.7) should fail. It was shown  = M ∗ . As condiabove that equation (3.8) holds with M tion (3.4) does not hold for the ith user, equation (3.9), and hence (3.7) fails. Thus, from (3.2) power level of the ith user is zero, pi = 0, at the equilibrium. As this holds for any  + 1, . . . , M}, the equilibrium power levels for these i ∈ {M users are zero. We now argue that the given boundary solution is unique. One possibility is the existence of an ith user, where 1  i  M ∗ , to have zero power. This cannot be a Nash  = M ∗. equilibrium, as it follows from (3.8) and (3.9) with M Another possibility is the existence of an ith user, where M ∗  i  M, transmitting with positive power level. This cannot be an equilibrium, either, as it was shown above that (3.7) fails for such an ith user, and pi = 0 follows directly from the reaction function (3.2). All possible boundary solutions can be captured by various combinations of these two cases. Consider the case where a subset of users among 1, . . . , M ∗ use zero power whereas some of the users among M ∗ + 1, . . . , M use positive power levels. Since for the subset of users with positive power levels among M ∗ + 1, . . . , M condition (3.4) does not hold, they cannot be in equilibrium following an argument similar to the one above. Otherwise, as condition (3.4) holds for the subset of users with zero power level among 1, . . . , M ∗ , they cannot be in equilibrium, either. We conclude, therefore, that the boundary solution is unique. Finally, in the case where no M ∗ exists satisfying condition (3.4), all users fail to satisfy (3.4), and the only solution is the trivial one, pi∗ = 0 ∀i. 

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4. Update schemes for mobiles, and stability

Thus, we obtain

In this section, we investigate the stability of the Nash equilibrium in the given model under two relevant asynchronous update schemes: parallel and random update. We establish a sufficient condition which guarantees the convergence to the unique equilibrium point for both algorithms. 4.1. Parallel Update Algorithm (PUA) In the PUA, users optimize their power levels at each iteration (in discrete time intervals) using the reaction function (3.2). If the time intervals are chosen to be longer than twice the maximum delay in the transmission of power level information, it is possible to model the system as a delay-free one. In a system with delays, there are subsets of users, updating their power levels given the delayed information. The algorithm is given by     (n)  1  (n) 1 (n+1) pi ai − , = i y −i , ai = max 0, yj hi L j =i

yj(n)

= hj pj(n) ,

(4.1)

or equivalently by yi(n+1)

  1  (n) = max 0, ai − yj , L j =i

whose global stability is established in the next theorem. This means that PUA converges to the unique Nash equilibrium of theorem 3.1 given as    1 1 pi∗ = max 0, hj pj∗ ai − (4.2) hi L j =i

from any feasible initial point, pi
0 ∀i. Theorem 4.1. PUA is globally stable, and converges to the unique equilibrium solution from any feasible starting point if the following condition is satisfied: M −1 < 1. L

(4.3)

Proof. Let us define the distance between the ith user’s power level received in the base station at any time (n) and received equilibrium power level as yi(n) := yi(n) − yi∗ . We consider first the case when yi∗ > 0 for an arbitrary ith user. Then, given the received power levels of all users except the (n) ith one at time n, yj j = i, we have the following from (4.1) and (4.2):    y (n)  1  (n)   −i  < , y < , if a  i j  L  (n+1)   L  j =  i y    i  1  (n)    = y else.  j ,  L j =i

 (n)   (n+1)   1 y . y i j L

(4.4)

j =i

Next, we consider the case when the received equilibrium power level for an arbitrary ith user is zero, yi∗ = 0. Then, from (4.1) and (4.2) it follows that    (n) y −i  1  (n)   (n+1)    , if a , y > i j  y   L L i   j =i 0, else. Thus, the inequality (4.4) holds for any ith user at any time instant n for both cases. We now show that (4.3) is a sufficient condition for the right-hand side of (4.4) to be a contraction mapping. Let y∞ denote the l∞ -norm of the vector (y1 y2 . . . yM )T , i.e., y∞ = max |yi |. i

(4.5)

Then, from (4.4),  (n)  M − 1   (n+1)   y   1 max y   y (n)  . j ∞ ∞ L i L j =i

Hence, (4.4) is a contraction mapping under condition (4.3), which leads to the stability and global convergence of the PUA (4.1).  We finally note that using an initial admission control mechanism and user dropping scheme, which limits the number, M, of users in the cell, this condition can easily be satisfied for a given L. Thus, the stability and convergence of the algorithm follows. 4.2. Random Update Algorithm (RUA) Random update scheme is a stochastic modification of PUA. The users optimize their power levels in discrete time intervals and infinitely often, with a predefined probability 0 < πi < 1. Thus, at each iteration a set of randomly picked users among the M update their power levels. The system with delay is also similar to PUA. The users make decisions based on delayed information at the updates, if the round trip delay is longer than the discrete time interval. The RUA algorithm is described by    hi i y (n) , ai , with probability πi , −i (n+1) = yi y (n) , with probability 1 − πi , i where i was defined in (4.1). We already know from the proof of theorem 4.1 that if user i updates, then (4.4) holds. Hence, for each i = 1, . . . , M,   E yi(n+1)     = E yi(n+1) user i updates at time n πi    (n)  + E yi user i does not update at time n (1 − πi )  (n)  πi   (n)  E yj + (1 − πi )E yi .  (4.6) L j =i

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Using again the l∞ -norm defined in (4.5), but with |yi | replaced by E|yi | (that is, y∞ := maxi E|yi |), and following steps similar to the ones of PUA, we obtain  (n+1)   max E y i

i

   M − 1 y (n)  max πi + max(1 − πi )y (n)   ∞ ∞ i i L

which leads to  (n+1)  y   ∞



   M −1 π + (1 − π ) y (n) ∞ , L

where π < 1 and π > 0 are the upper and lower limits for the update probability of the ith user respectively, π < πi < π. Therefore, M −1 π + (1 − π ) < 1 L

(4.7)

is a sufficient condition for the right-hand side of (4.6) to be a contraction mapping, and for the stability and convergence of RUA in norm. We also note that when all users have the same update probability, πi = π ∀i, this condition simplifies to (M − 1)/L < 1, same sufficient condition (4.3) as the one for PUA. We show next a stronger result, almost sure (a.s.) convergence of RUA, under condition (4.7). By the Markov inequality and using the definition of the l∞ -norm, we have ∞ ∞      E|yi(n) | P yi(n)  > ε  ε n=1



n=1 ∞ 

1 ε

 (n)  y  , ∞

(4.8)

n=1

where P (·) denotes the underlying probability measure. (n) Since E|yi | is a contracting sequence with respect to the l∞ -norm as shown,   (n)   y   α y (n−1)  ∞  ∞(0) n  · · ·  α y ∞ , where 0 < α < 1. Using this in (4.8), it follows that ∞ ∞      y (0) ∞ P yi(n)  > ε  αn ε n=1

n=1

=

y (0) ∞ , ε(1 − α)

and thus, follows that ∞   (n)   P y  > ε  i

n=1

K , ε(1 − α)

where K is a constant (actually, K = y (0)∞ ). Hence, the
(n) increasing sequence of partial sums N n=1 P (|yi | > ε) is bounded above by K/(ε(1 − α)). Thus, it converges for every

ε > 0. From the Borel–Cantelli lemma [2,3], it then follows that    (n)   P lim sup ω: y  > ε = 0. n→∞

i

Hence, RUA converges also a.s. under condition (4.7). 4.3. Comparison of PUA and RUA One important feature of PUA is that it ascribes a myopic behavior to the users, that is they optimize their power levels based on instantaneous costs and parameters, ignoring future implications of their actions. This behavior of users is realistic for the analyzed wireless network as it may not be feasible or even possible for a mobile to estimate future values of total interference in the cell or future variations in its own channel gain. In the case of RUA, the users are again myopic and update their power levels based on instantaneous parameters. But, not all of them act at every iteration; whether a particular user responds or not is determined probabilistically. In the limiting case when all update probabilities, πi , are equal to 1, RUA is the same as PUA. An advantage of RUA, however, is that through it one can investigate the convergence of theproposed scheme when there are random delays in the system. Such delays may be due to differences in the processing or propagation times. As we will see in the simulations included in section 6, in a delay-free system if all the users have the same initial power level, then RUA performs better than PUA. This is due to the myopic behavior of users, as well as the inherent randomization in the case of RUA. On the other hand, the opposite is true for a system with delay as variations in delay provide sufficient randomization, and PUA becomes more advantageous due to frequent updates. More detailed observations on the convergence of both algorithms can be found in section 6 for both delay-free and delayed cases. 5. Pricing strategies at the base station In a noncooperative network, pricing is an important design tool as it creates an incentive for the users to adjust their strategies, in this case power levels, in line with the goals of the network. In the CDMA system we are studying here, the price per unit power of the ith user, λi , is determined by the base station in a manner to be discussed shortly. We introduce a pricing scheme where the price charged to each user is proportional to the received power from the user at the base station. Thus, the price is proportional to the channel gain of the ith user, λi = ki hi . The inner Nash solution by itself does not guarantee that the users with nonzero power levels will meet the minimum SIR requirement to establish a connection to the base station. Achieving the necessary SIR level is obviously crucial to the successful operation of the system. Furthermore, one has to recognize that different communication applications in wireless systems leads to different types of users and SIR requirements in addition to the minimum SIR level.

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In view of these considerations, we will consider in this section two different pricing schemes: (i) A centralized pricing scheme. Users are divided into classes, with all users belonging to a particular class having the same utility function parameter (ui ). Further, all users within a class have the same SIR requirement. The role of the base station is to set prices for these different classes such that, under the resulting Nash equilibrium, the SIR targets of the users are met. (ii) Decentralized, market-based pricing. The base station sets a single price for all users, and the users choose their willingness to pay parameter, ui , to satisfy their QoS requirements. As compared to the centralized scheme, this one is more flexible, and allows users to compete for the system resources by adjusting their individual ui ’s. 5.1. Centralized pricing schemes and admission control First, we consider the symmetric-user case where every mobile has the same SIR requirement, and for convenience we let ui = 1. It is possible to find a simple pricing strategy by picking the price directly proportional to the channel gain, λi = khi , where the pricing factor, k, is user independent. The parameter k is a function of the number of users and the desired SIR level. Notice that this approach is equivalent to centralized power control as the prices are adjusted by the base station in such a way that the mobiles use the power levels determined by the unique Nash equilibrium as a result of their individual optimization. Moreover, the base station can set the prices such that the SIR requirements of the users are satisfied. A precise result covering this case is now captured by the following theorem. Theorem 5.1. Assume that the users are symmetric in their utilities, ui = 1 ∀i, they have the same minimum SIR requirement, γ ∗ , and are charged in proportion with their channel gain, λi = khi . Then the maximum number of users, M ∗ , the system can accomodate is bounded by M∗
M ∗ , all users fall below the desired SIR level (γ ∗ ) due to the symmetry. In this case, dropping some of the users from the system in order to decrease the number of users M below the threshold (M ∗ ) would lead to a viable solution. Next, we consider the case where the network may provide multiple service levels and multiple pricing schemes. For this more general case, it is convenient to split the mobiles in a cell into multiple groups according to their need for bandwidth, or in our context, their desired SIR levels, where the users within each group are symmetric. Using a multiple pricing scheme, a solution capturing multiple user groups can be obtained straightforwardly. 5.2. Market-based scheme It is natural to think of each user within a cell having different SIR requirements, which can be quantified with the userspecific utility parameter u. The base station can implement a natural pricing strategy by formulating the pricing parameter directly proportional to the channel gain, λi = khi . However, it is impossible in this case for the base station to calculate the parameter k, as the user preferences are unknown to the base station. Hence, after the base station sets an appropriate value for price (k), each user dynamically updates its power level by minimizing its cost under parallel update (PUA) or random update (RUA) algorithms. As a result, a distributed and market-based power control scheme is obtained. Due to the interference in the CDMA system, each user affects others. Hence, the ith mobile can adjust its utility parameter, ui , dynamically according to its minimum SIR level, γi∗ , given the interference at the base station. From (3.2) and (2.2), it follows that ui >

  λi  ∗ γi + 1 y −i + σ 2 . Lhi

The parameter ui is bounded below by the total received power at the base station. This can be interpreted as follows. If a mobile is in a cell where the interference is low, the mobile can achieve the desired SIR level with a low power, hence paying a lower price. However, in a situation where many users compete for the SIR, the mobile has to use more power, and pay a higher price to reach the same SIR level. In the latter case, the user’s willingness to spend more can be justified with a higher ui based on (3.2). We note that, together with the utility function, the utility parameter ui quantifies the user’s desire for the SIR. The base station can limit aggressive requests for SIR even in the case when a user pays for its excessive usage of power, by setting an upper-limit, ymax , for the received power of the ith user at the base station: yi  ymax . Hence, unresponsive users can be punished by the base station in order to preserve network resources. From (3.2), we can obtain an upper-bound

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on the value of ui . Furthermore, this bound depends only on user-independent parameters, such as the upper limit of the total received power at the based station, maximum number of mobiles, Mmax , and the spreading gain, L, if proportional pricing is used: ui 

" k! 2 σ + (L + Mmax − 1)ymax ∀i. L

When this bound is combined with a simple admission control scheme, limiting the number of mobiles to Mmax , the base station can provide guarantees for a minimum SIR level γmin : γmin =

Lymax . (Mmax − 1)ymax + σ 2

A tradeoff is observed in the choice of the design parameters γmin versus Mmax . If the network wants to provide guarantees for a high SIR level, then it has to make a sacrifice by limiting the number of users. In addition, users may implement a distributed admission scheme according to their budget constraints and desired SIR levels. If the price necessary to reach a SIR level exceeds the budget, Bi , of the user, that is kγi∗  L

 y −i + σ 2
Bi ,

Figure 2. Comparison of power and SIR final values of the mobiles for the fixed and proportional pricing schemes.

information flow to users. Subsequently, delay is introduced to the system to make the setting more realistic. 6.1. Effect of the pricing parameters

then the user may simply choose not to transmit at all.

6. Simulation studies The proposed power control scheme has been simulated numerically using MATLAB. Here, we first investigate different pricing schemes for symmetric mobiles in the fixed-utility case. Then, we analyze the robustness of the system under varying parameters such as noise, the number of users, and channel gains. Furthermore, the rate of convergence of both update schemes, PUA and RUA, are studied both in the delayfree and delayed cases. Finally, we investigate a system consisting of users with various utility parameters. All results of the simulations are valid for both update schemes, PUA and RUA, where the only difference between the two is the convergence rate. Simulation parameters are chosen as follows, unless otherwise stated. The spreading gain L = 128 is chosen in accordance with IS-95 standard [9]. Noise factor is σ 2 = 1 whereas the stopping criterion or distance to equilibrium is given by ε = 10−5 . The users are assumed to be located randomly in the cell where the distance of the ith user to the base station, di , is uniformly distributed between d0 = 10 and dmax = 100. The channel gains of users are determined by a simple largescale path loss formula hi = (d0 /di )2 where the path loss exponent is chosen as 2 corresponding to open air path loss. Under the fixed-utility case, users have the same utility parameter, ui = 1 ∀i. The initial condition for simulations is pi = 1 ∀i, an estimated value for establishing initial communication between the mobile and the base station. In the simulations, a discrete time scale is used. The delay-free case is characterized by a time span that is long enough for perfect

In the first simulation, proportional and fixed pricing schemes are compared. For simplicity, we first choose the users being symmetric under both fixed pricing, λi = λ, and proportional pricing, λi = khi . For illustrative purposes the number of users is chosen small, M = 20. In figure 2, the equilibrium power and the SIR values of each user can be seen under both pricing schemes. In the top graph, power values of the users with different channel gains are almost the same under fixed pricing. Hence, the users with lower channel gains achieve lower SIR values. In contrast, all users meet a minimum SIR level under proportional pricing, regardless of their channel gain. An intuitive explanation is that under proportional pricing the distant users are allowed to use more power to attain the necessary SIR level. We also note that proportional pricing is “fair” in the sense that the users are not affected by their distance to the base station. Convergence of users’ power levels to their equilibrium values is demonstrated in figure 3(a) under PUA, and figure 3(b) under RUA with update probability being 0.6. In both cases there are 10 users and L = 20. The effect of pricing is investigated in the next simulation for a single class of users by varying the pricing parameter, k, under proportional pricing. Equivalently, this simulation can be interpreted as varying the utility parameter, u. Both parameters play a crucial role in the system by affecting the overall power and SIR levels. From (3.2), the effect of ui on the system is inversely proportional to ki . In figure 4 it can be observed that a gradual increase in k from 1 to 4, i.e., an increase in price, affects the system in a such a way that both power and SIR values decrease. Since, with an increase in the price, the users decrease their powers to the same extent leading to lower SIR values given a constant noise level. Equivalently,

CDMA UPLINK POWER CONTROL AS A NONCOOPERATIVE GAME

(a)

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(b)

Figure 3. Convergence of users’ individual power levels to the equilibrium values versus number of iterations under (a) PUA and (b) RUA with π = 0.6.

Figure 4. Effect of the pricing parameter k (utility parameter 1/u) on the SIR and the power levels of users.

a decrease in u, the users’ level of request for SIR, gives the same result. Furthermore, the observations match theoretical calculations for the single class case in accordance with (5.2). 6.2. Convergence rate and robustness of algorithms 6.2.1. Simulations without delay The convergence rate of the two update schemes is of great importance, as it has a direct effect on the robustness of the system. We have simulated PUA and RUA for different numbers of symmetric users under a single pricing scheme. In figure 5, the number of iterations to the equilibrium point is shown for different probability values of RUA and also for PUA (which corresponds to RUA with the update probability equal to one). As the number of users increase the optimal update probability decreases. This result is in accordance with

Figure 5. Convergence rate for different update probabilities and increasing numbers of users. L = 800.

the one in [6] where it is shown that in a quadratic system without delay, an approximate value for the optimal update probability is 2/3, as the number of users goes to infinity. On the other hand, if the number of users is much smaller than the spreading gain, M  L, then PUA is superior to RUA. Next, we investigate the robustness of the system in the delay-free case. First, we analyze it under increasing noise, σ 2 . The background noise is increased step by step up to 100% of its initial value. Accordingly, the base station allows users to increase their powers by decreasing the prices by the same percentage in the fixed-utility case. The simulation is repeated with N = 20 users under a proportional pricing scheme. We observe in figure 6(a) that the power values increase in response to the increasing noise to keep the initial SIR constant. Similarly, we increase the number of mobiles in the system threefold in figure 6(b). It has the same effect as increasing the noise due to the nature of CDMA. Again by

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(a)

(b)

Figure 6. Power and SIR final values for increasing noise (a) and numbers of users (b).

adjusting the prices accordingly, all users keep their SIR levels. Same results are obtained equivalently under the marketbased pricing scheme, where users adjust their utility parameter, u, dynamically while the pricing parameter determined by the base station is kept constant. As a conclusion, these observations confirm the robustness of the proposed power control scheme. Finally, we simulate the system in a realistic setting under a single pricing scheme. The number of users, N = 10 taken as initial value, is modeled as a Markov chain. Arrival of new mobiles is chosen to be Poisson with an average of 2 new users per time interval. Call durations are exponentially distributed with an average of 20 time intervals. We observe the average percentage difference between the theoretical equilibrium and the current operating point of the system in terms of power values of users for some period of time. In the simulation, PUA is chosen as the update algorithm. The initial condition is the equilibrium point for users. The simulated system operates within 1% range of the equilibrium points, and the results are very similar to those of in figure 7. Heretofore, robustness of the system was investigated for static mobiles. The movements of the users within the cell result in varying channel gains. In the next simulation, the locations, hence, channel gains of users are varied randomly. The movement of ith user is modelled after a random walk where a random value is added to the distance of the user to the base station di at each time instant. Hence, we obtain di (n + 1) = di (n) + x(n) where x(n) ∈ [−1, 1] is uniformly distributed. Furthermore, the setting used in this simulation is the same as previous one. From figure 7, the system again operates within 1% distance to equilibrium. 6.2.2. Simulations with delay We introduce the delay factor into the system in the following way: users are divided into d equal size groups, and each group has an increasing number of units of delay. First, the convergence rates of the two update schemes are compared

Figure 7. Average percentage distance to the equilibrium point versus time. Channel gains, hi , are varied based on random movements of users.

and contrasted under delay-free and delayed conditions. The update probability of RUA is chosen as 0.66 which corresponds to the optimal update probability for a large number of users. In the delay-free case RUA outperforms PUA as the number of users increases. In the delayed case, however, PUA is always superior to RUA. Then, the simulation investigating the convergence rate of RUA for various update probabilities is repeated in the delayed case. The result shown in figure 8 is different from the previous one in figure 5. Here, PUA converges faster than RUA for any number of users verifying previous results. Finally, a market-based pricing scheme with proportional pricing at the base station, k = 1, is investigated. There are two groups of users, which are symmetric within themselves. Users in each group have different utility parameters, u. The group with higher u is labeled as the “priority” user group, while the other one is called the “regular” user group. In or-

CDMA UPLINK POWER CONTROL AS A NONCOOPERATIVE GAME

Figure 8. Convergence rate for different update probabilities and increasing numbers of users (with delay). L = 800.

Figure 9. SIR and power levels at the base station (or prices) of two selected users from priority and regular user groups versus time.

der to observe the effect of varying number of users on the SIR levels, we let a sample user from each group make a long enough call. At the same time, the number of users in each group and channel gains of the users are varied similar to those in previous robustness simulations to create realistic disturbances in the system. For simplicity, the values of the utility parameters are kept constant throughout the simulation. In figure 9, it is observed that a priority user always obtains a higher SIR than a regular user. Another observation is that priority users use a higher power level, and therefore pay more than regular users, as expected. The fluctuation in the power levels is due to the varying number of users, and varying total demand for SIR in the system. 7. Conclusion In this paper, we have developed a mathematical model within the framework of noncooperative game theory, and have ob-

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tained distributed, asynchronous control mechanisms for the uplink power control problem in a single cell CDMA wireless network. Existence of a unique Nash equilibrium has been proven, and convergence properties of parallel and random update schemes have been investigated analytically and numerically. Moreover, conditions for the stability of the unique equilibrium point under the update algorithms have been obtained and analyzed accordingly. We have shown that the unique Nash equilibrium has the property that, depending on the parameter values, only a subset of the total number of mobiles are active. Some of the users are dropped from the system as a result of the power optimization. By defining a utility function and a utility parameter, user requests for SIR were modeled dynamically. Furthermore, the relationship between the SIR level of the users and the pricing has been investigated for two different pricing schemes for the fixed and varying utility cases. It has been shown both analytically and through simulations that choosing an appropriate pricing strategy guarantees meeting the minimum desired SIR levels for the active users in the fixed-utility case. In addition, the principles for an admission scheme have been investigated under the market-based scheme. The results obtained indicate that the proposed framework provides a satisfactory decentralized and market-based solution. The algorithms in this model are practically implementable, as the only information a user requires to update its power other than own preferences and fixed parameters is the total received power level from the base station. This information may be conveyed incrementally to reduce the overhead in the case of frequent updates. Although a specific cost structure is chosen in this paper, most of the results may be extended to more general cost functions. Another possible extension to this work is to a multiple cells model, where the effect of neighboring cells are taken into account. A further topic of interest is the development of the counterparts of the results in the case of multiple base stations, which brings up the challenging issue of handoff. Acknowledgements Research supported in part by grants NSF ANI 98-13710, NSF INT 98-04950, NSF CCR 00-85917, AFOSR MURI AF DC 5-36128, and ARMY OSP 35352-6086. References [1] T. Ba¸sar and G.J. Olsder, Dynamic Noncooperative Game Theory, 2nd ed. (SIAM, Philadelphia, PA, 1999). [2] P. Billingsley, Probability and Measure, 2nd ed. (Wiley, New York, 1986). [3] J.L. Doob, Stochastic Processes (Wiley, New York, 1953). [4] D. Falomari, N. Mandayam and D. Goodman, A new framework for power control in wireless data networks: Games utility and pricing, in: Proc. Allerton Conf. on Communication, Control, and Computing, Illinois, USA (September 1998) pp. 546–555. [5] H. Ji and C. Huang, Non-cooperative uplink power control in cellular radio systems, Wireless Networks 4(3) (April 1998) 233–240.

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[6] R.T. Maheswaran and T. Ba¸sar, Multi-user flow control as a Nash game: Performance of various algorithms, in: Proc. 37th IEEE Conf. on Decision and Control (December 1998) pp. 1090–1095. [7] D. Mitra and J. Morrison, A novel distributed power control algorithm for classes of service in cellular cellular CDMA networks, in: Proc. 6th WINLAB Workshop on 3rd Generation Wireless Information Networks, New Jersey, USA (March 1997). [8] L.L. Peterson and B.S. Davie, Computer Networks: A System Approach, 2nd ed. (Morgan Kaufmann, San Francisco, CA, 2000). [9] T.S. Rapaport, Wireless Communications: Principles and Practice (Prentice Hall, Upper Saddle River, NJ, 1996). [10] C.U. Saraydar, N. Mandayam and D. Goodman, Pareto efficiency of pricing-based power control in wireless data networks, in: WCNC (1999). [11] C.W. Sung and W.S. Wong, Power control for multirate multimedia CDMA systems, in: Proc. of IEEE INFOCOM, Vol. 2 (1999) pp. 957–964. [12] A.J. Viterbi, CDMA Principles of Spread Spectrum Communication (Addison-Wesley, Reading, MA, 1995). [13] R.D. Yates, A framework for uplink power control in cellular radio systems, IEEE Journal on Selected Areas in Communications 13 1341–1347 (September 1995) 1341–1347.

Tansu Alpcan received the B.S. degree in electrical engineering from Bo˘gaziçi University, Istanbul, Turkey, in 1998, and the M.S. degree in electrical and computer engineering from the University of Illinois at Urbana-Champaign (UIUC) in 2001. His research interests include game theory, and control and optimization of wireline and wireless communication networks. He is the recipient of a Fulbright scholarship in 1999, and is a student member of IEEE. Currently, he is a Ph.D. candidate in electrical and computer engineering at UIUC. E-mail: [email protected]

Tamer Ba¸sar received B.S.E.E. degree from Robert College, Istanbul, and M.S., M.Phil., and Ph.D. degrees in engineering and applied science from Yale University. After stints at Harvard University and Marmara Research Institute (Gebze, Turkey), he joined the University of Illinois at UrbanaChampaign in 1981, where he is currently the Fredric G. and Elizabeth H. Nearing Professor of Electrical and Computer Engineering. He has published extensively in systems, control, communications, and dynamic games, and has current interests in robust nonlinear and adaptive control, modeling and control of communication networks, con-

ALPCAN ET AL.

trol over wireless links, resource management and pricing in networks, risksensitive estimation and control, and robust identification. Dr. Ba¸sar is currently the Deputy Editor-in-Chief of Automatica, Editor of the Birkhäuser Series on Systems & Control, Managing Editor of the Annals of the Society of Dynamic Games, and a member of editorial and advisory boards of several international journals. He has received several awards and recognitions over the years, among which are the Medal of Science of Turkey (1993), Distinguished Member Award of the IEEE Control Systems Society (CSS) (1993), and the Axelby Outstanding Paper Award of the same society (1995). He is a member of the National Academy of Engineering, Fellow of IEEE, and a past president of CSS. E-mail: [email protected]

R. Srikant received his B. Tech. from the Indian Institute of Technology, Madras, India, in 1985, and the M.S. and Ph.D. degrees from the University of Illinois in 1988 and 1991, respectively, all in electrical engineering. He was a Member of Technical Staff in AT&T Bell Laboratories from 1991 to 1995. Since August 1995, he has been with the University of Illinois where he is currently an Associate Professor in the Department of General Engineering and Coordinated Science Laboratory, and an Affiliate in the Department of Electrical and Computer Engineering. He is the Chair of the 2002 IEEE Computer Communications Workshop, and serves as an associate editor of Automatica and IEEE/ACM Transactions on Networking. His research interests include communication networks, queueing theory, information theory and stochastic control. He received an NSF CAREER award in 1997. E-mail: [email protected]

Eitan Altman received the B.Sc. degree in electrical engineering (1984), the B.A. degree in physics (1984) and the Ph.D. degree in electrical engineering (1990), all from the Technion-Israel Institute, Haifa. In 1990, he further received his B.Mus. degree in music composition in Tel-Aviv University. Since 1990, he has been with INRIA (National Research Institute in Informatics and Control) in Sophia-Antipolis, France. He has been on sabbatical since 2001 in CESIMO, Facultad de Ingenieria, Universidad de Los Andes, Merida, Venezuela. His current research interests include performance evaluation and control of telecommunication networks, stochastic control and dynamic games. In recent years, he has applied control theoretical techniques in several joint projects with the French telecommunications company – France Telecom. E-mail: [email protected]